I had gotten US citizenship 5 years back but never returned my Nepali passport as I should have. Now with new constitution I hear one can get dual citizenship. Am I allowed to simply renew my old passport?

I think you can, though I might be missing something. The only thing is "you cannot vote" in Nepal, when u have US citizenship. Again, I am not an expert, I just think that is the case.

Its not the dual citizenship that's approved in the new constitution, its the NRN status; it will allow you to have social and economic association with Nepal but will not allow you to be politically involved, neither will it provide you the voting rights.

What about the passport ? Can we keep US passport as well as Nepali passport ?

What I have understood is its only like NRN card saying "Non Resident Citizen" having non-political rights but won't have passport.

You cannot keep the passport; you are obligated to have your Nepali passport cancelled once you acquire your US passport. You cannot travel with your Nepali passport.

It's NRN why do you want to be dual citizen if don't be want Nepali citizen.

Your old Nepali citizenship gets voided as soon as you have acquired citizenship of a different country.

Under new constitution, you can apply for a citizenship specifically designated for those that have acquired citizenships of countries outside SAARC region. 

This will allow you to enjoy all the rights of a Nepali citizen except being able to get politically involved (no voting).

But unlike NRN card, there won't be any economic restrictions. You get to enjoy all other social and economic rights (purchasing land and property like an average Nepali) under new constitution. 

This should also allow you to apply/acquire a Nepali
passport and a dual citizen should no longer be required to have a visa to live and work in Nepal.

This is just a constitution....wait till we get laws and regulations, codes etc till you interpret it. As of now only Supreme Court can interpret what the language of constitution means.

But based on general understanding nrn are getting one step down from naturalized citizen, which includes social, economic and religious right. Now I don't know what rights are under social, economic and religious right. Passport I am not sure, and personally I don't care about passport.

Dual nationality is not permitted under Nepal law. Follow this link, it may help you.
http://www.merokalam.com/dual-citizenship-nepal/